Symmetric Group Character Degrees

• Symmetric group character degrees are invariants, computed through the hook-length formula, that uniquely characterize the structure of Sₙ.
• The analysis combines combinatorial partition theory with Molien's theorem to confirm group isomorphism when degree patterns match.
• Applications extend to finite group classification, representation theory, and algorithmic recognition of group algebra structures.

The symmetric group character degrees are an essential invariant in the representation theory of finite groups, encapsulating both profound combinatorial geometry and robust algebraic structure. For the symmetric group $S_n$, the set of complex irreducible character degrees—parameterized by partitions of $n$ through the hook‐length formula—uniquely determines the group up to isomorphism. This interplay between degrees, combinatorics, and group structure provides not only a window into the architecture of $S_n$ itself, but also serves as a testing ground for broader conjectures in group theory and representation theory.

1. Character Degree Patterns and Group Determination

The multiset of irreducible character degrees, specifically the first column $X_1(S_n)$ of the character table, fully determines the symmetric group among all finite groups. That is, if $G$ is a finite group and $X_1(G) = X_1(S_n)$ (meaning $G$ has exactly the same degrees, with multiplicities, as $S_n$), then $G \cong S_n$ (Tong-Viet, 2011). Key formulas from the character table encode vital group invariants:

• The total order: $|G| = \sum_{\chi \in \mathrm{Irr}(G)} (\chi(1))^2$.
• Minimal degrees: For $S_n$ with $n \geq 15$, the minimal nontrivial degree is $d_1(S_n) = n - 1$, and the second minimal nontrivial degree is $d_2(S_n) = n(n - 3)/2$ (via Rasala's results).

Notably, the degree pattern, together with the number of conjugacy classes and the derived subgroup index $|G : G'|$, suffices to enforce that any such $G$ is perfect and has $|G| = n!$, leading to $G \cong S_n$ as soon as the degree pattern matches.

2. Mathematical Formulas for Character Degrees

The irreducible degrees $\chi^\lambda(1)$ for $S_n$ are computed through the hook length formula: $$\chi^\lambda(1) = \frac{n!}{\prod_{(i,j)\in Y_\lambda} h(i,j)}$$ where $h(i,j)$ is the hook length of the node $(i,j)$ in the Young diagram labeled by partition $\lambda$. The degree pattern is denoted by $X_1(S_n) = (d_1, d_2, \ldots, d_k)$, organized according to partitions. For large $n$, the minimal degrees admit uniform computation, while higher degrees require analysis of local Young diagram combinatorics.

Molien's Theorem provides a linkage between character degrees and the complex group algebra: $$\mathbb{C}G_1 \cong \mathbb{C}G_2 \quad \Longleftrightarrow \quad X_1(G_1) = X_1(G_2)$$ indicating that the full structure of $\mathbb{C}S_n$ is captured by its character degrees.

3. Uniqueness Criterion via the Complex Group Algebra

The complex group algebra $\mathbb{C}S_n$ is an invariant that "remembers" the symmetric group. If a finite group $G$ satisfies $\mathbb{C}G \cong \mathbb{C}S_n$, then necessarily $G \cong S_n$. This result prescribes a rigid correspondence between the algebraic structure of the group algebra and the underlying group itself. The uniqueness is underpinned by Molien's Theorem—any group with the same character degree pattern as $S_n$ must present the same group algebra decomposition, enforcing isomorphism.

4. Classification Results and the Huppert Conjecture

The character degree pattern of $S_n$ is leveraged to support and extend conjectures such as Huppert's, which posits that non-abelian simple groups are uniquely determined by their sets of character degrees. The methodology in (Tong-Viet, 2011)—which applies invariant analysis and classification results to the symmetric groups—can be adapted to other infinite families such as alternating or sporadic groups, suggesting utility for resolving analogous problems throughout finite group theory.

5. Applications and Structural Implications

The detailed correspondence between character degrees and group structure has numerous applications:

• It provides a strong invariant for group classification, enabling the detection and distinction of symmetric groups within larger families.
• In representation theory, the explicit control over character degrees facilitates analysis of $S_n$ modules, decomposition numbers, and block theory.
• The interplay is instrumental in algebraic combinatorics, where symmetric groups act as symmetry controllers for combinatorial objects.
• The structure of $\mathbb{C}S_n$ as reconstructed from degrees is used in testing recognition algorithms for group algebras.
• The result also impacts the paper of the arithmetic spectrum of group representations and the detection of "hidden" group-theoretic structure from local data.

6. Broader Context, Consequences, and Open Directions

These findings resolve the identification question posed in 2, Question 126—no two non-isomorphic finite groups share the character degree pattern of $S_n$. The approach demonstrates a general principle: for certain "natural" finite groups, character degrees are categorical. The techniques engage combinatorial partition theory, the theory of Young diagrams, and deep group algebraic results, forming a template for further studies in the classification of groups via their representation-theoretic invariants.

A plausible implication is that characteristic-free invariants such as character degrees may play a decisive role in algorithmic and computational group recognition tasks, and may inform the development of new algorithms for automorphism group identification, cohomology module decomposition, and block-wise analysis in modular representation theory.

7. Key Theorems and Formulas Table

Invariant	Formula/Result	Context
Order from degree pattern	$|G| = \sum_{\chi \in \mathrm{Irr}(G)} (\chi(1))^2$	Holds for all finite $G$
Character degree (hook formula)	$$\chi^\lambda(1) = n!/\prod_{h \in \mathrm{hook}(\lambda)} h$$	$\lambda$ a partition
Minimal degrees for $S_n$	$d_1(S_n) = n-1$, $d_2(S_n) = n(n-3)/2$	$n \geq 15$
Molien's theorem	$\mathbb{C}G_1 \cong \mathbb{C}G_2$ iff $X_1(G_1)=X_1(G_2)$	Character algebra

The table summarizes the principal formulas utilized in the determination of symmetric group structure from character degrees. These analytic tools, together with the cited theorems, form the backbone of the group recognition results for $S_n$.

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This comprehensive linkage between character degrees and group structure for symmetric groups not only answers longstanding classification questions, but also establishes a paradigm for the utility of degree patterns as diagnostic invariants in finite group theory and representation theory.